Past

A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced by C. Chevalier and F Debbasch, both in a heuristic and analytic way.  Roughly speaking, they are characterised by the existence at each (proper) time (of the moving particle) of a (local) rest frame where the random part of the acceleration of the particle (computed using the time of the rest frame) is brownian in any spacelike direction of the frame.

I will explain how the tools of stochastic calculus enable us to give a concise and elegant description of these random paths on any Lorentzian manifiold.  A mathematically clear definition of the the one-particle distribution function of the dynamics will emerge from this definition, and whose main property will be explained.  This will enable me to obtain a general H-theorem and to shed some light on links between probablistic notions and the large scale structure of the manifold.

All necessary tools from stochastic calculus and geometry will be explained.

Abstract:  Modern techniques for computing multi-particle and multi-loop scattering amplitudes rely on a sophisticated use of on-shell recursion relations and generalised unitarity methods. I will show that these methods are ideally suited to interpretation in twistor space, where superconformal properties become manifest. In fact, the recursion relations of Britto, Cachazo, Feng & Witten provide a clear framework for the twistor diagram program initiated in the 1970s.
Tree-level scattering amplitudes in N=4 SYM are now known to possess a Yangian symmetry, formed by combining the original PSU(2,2|4) superconformal invariance with a second "dual" copy. I will also discuss very recent work constructing scattering amplitudes in a twistor space in which this dual superconformal symmetry acts geometrically.

Soliton like structures called “stable droplets” are found to exist within a paradigm reaction
diffusion model which can be used to describe the patterning in a number of fish species. It is
straightforward to analyse this phenomenon in the case when two non-zero stable steady states are
symmetric, however the asymmetric case is more challenging. We use a recently developed
perturbation technique to investigate the weakly asymmetric case.

We focus on structural models in corporate finance with roll-over debt structure and endogenous default triggered by limited liability equity-holders. We point out imprecisions and misstatements in the literature and provide a rationale for the endogenous default policy.

Inverse problems arise with regularity (sic!) in the context of our study of the deformation of solids, and its characterisation (in terms of diffraction and imaging) using radiation (neutrons and X-rays).

I wish to introduce several examples where the advancement of inverse problem methods can make a significant impact on applicatins.

1. Inverse eigenstrain analysis of residual stress states

2. Strain tomography

3. Strain image correlation

Depending on the time available, I may also mention (a) Rietveld refinement of diffraction patterns from polycrystalline aggregates, and
(b) Laue pattern indexing and energy dispersive detection for single grain strain analysis.

This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.

This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.

We propose a powerful prediction algorithm built upon Gaussian
processes (GPs). They are particularly useful for their flexibility,
facilitating accurate prediction even in the absence of strong physical models. GPs further allow us to work within a completely Bayesian framework. As such, we show how the hyperparameters of our system can be marginalised by use of Bayesian Monte Carlo, a principled method of approximate integration. We employ the error bars of the GP's prediction as a means to select only the most informative observations to store. This allows us to introduce an iterative formulation of the GP to give a dynamic, on-line algorithm. We also show how our error bars can be used to perform active data selection, allowing the GP to select where and when it should next take a measurement.

We demonstrate how our methods can be applied to multi-sensor prediction problems where data may be missing, delayed and/or correlated. In particular, we present a real network of weather sensors as a testbed for our algorithm.

The study of representation growth of infinite groups asks how the

numbers of (suitable equivalence classes of) irreducible n-dimensional

representations of a given group behave as n tends to infinity. Recent

works in this young subject area have exhibited interesting arithmetic

and analytical properties of these sequences, often in the context of

semi-simple arithmetic groups.

In my talk I will present results on the representation growth of some

classes of finitely generated nilpotent groups. They draw on methods

from the theory of zeta functions of groups, the (Kirillov-Howe)

coadjoint orbit formalism for nilpotent groups, and the combinatorics

of (finite) Coxeter groups.

The Kerr solutions to Einstein's equations describe rotating black holes. For the wave equation in flat-space and outside the non-rotating, Schwarzschild black holes, one method for proving decay is the vector-field method, which uses the energy-momentum tensor and vector-fields. Outside the Schwarzschild black hole, a key intermediate step in proving decay involved proving a Morawetz estimate using a vector-field which pointed away from the photon sphere, where null geodesics orbit the black hole. Outside the Kerr black hole, the photon orbits have a more complicated structure. By using the hidden symmetry of Kerr, we can replace the Morawetz vector-field by a fifth-order operator which, in an appropriate sense, points away from the photon orbits. This allows us to prove the necessary Morawetz estimate. From this we can prove a decay estimate of almost $t^{-1}$ for fixed $r$ and the corresponding decay rates at the event horizon and null infinity. The major innovation in this result is that, by using the hidden symmetries with the energy-momentum, we can avoid taking Fourier tranforms in time.

This is joint work with Lars Andersson.

This work in collaboration with J. Casado-Díaz deals with the asymptotic behaviour of two-dimensional linear conduction problems for which the sequence of conductivity matrices is bounded from below but not necessarily from above. On the one hand, we prove an extension in dimension two of the classical div-curl lemma, which allows us to derive a H-convergence type result for any L¹-bounded sequence of conductivity matrices. On the other hand, we obtain a uniform convergence result satisfied by the minimisers of a sequence of two-dimensional diffusion energies. This implies the closure for the L²-strong topology of $\Gamma$-convergence of the sets of equicoercive diffusion energies without assuming any bound from above. A few counter-examples in dimension three, connected with the appearance of non-local effects, show the specificity of dimension two in the two previous compactness results.

Can investors with irrational beliefs be neglected as long as they are rational on average ? Does unbiased disagreement lead to trades that cancel out with no consequences on prices, as implicitly assumed by the traditional models ? We show in this paper that there is an important impact of unbiased disagreement on the behavior of financial markets, even though the pricing formulas are "on average" (over the states of the world) unchanged. In particular we obtain time varying, mean reverting and countercyclical (instead of constant in the standard model) market prices of risk, mean reverting and procyclical (instead of constant) risk free rates, decreasing (instead of flat) yield curves in the long run, possibly higher returns and higher risk premia in the long run (instead of a flat structure), momentum in stock returns in the short run, more variance on the state price density, time and state varying (instead of constant) risk sharing rules, as well as more important and procyclical trading volumes. These features seem consistent with the actual (or desirable) behavior of financial markets and only result from the introduction of unbiased disagreement.

Inverse problems are often ill-posed, with solutions that depend sensitively on data. Regularization of some form is often used to counteract this. I will describe an approach to regularization, based on a Bayesian formulation of the problem, which leads to a notion of well-posedness for inverse problems, at the level of probability measures.

The stability which results from this well-posedness may be used as the basis for understanding approximation of inverse problems in finite dimensional spaces. I will describe a theory which carries out this program.

The ideas will be illustrated with the classical inverse problem for the heat equation, and then applied to so more complicated inverse problems arising in data assimilation, such as determining the initial condition for the Navier-Stokes equation from observations.

I will present a new proof of Mumford's conjecture on the rational cohomology of moduli spaces of curves, which is substantially different from those given by Madsen--Weiss and Galatius--Madsen--Tillmann--Weiss: in particular, it makes no use of Harer--Ivanov stability for the homology of mapping class groups, which played a decisive role in the previously known proofs. This talk represents joint work with Soren Galatius.

Presheaves on categories crop up everywhere! In this talk, I'll give a

gentle introduction to 2-categories, and discuss the notion of a

presheaf on a 2-category. In particular, we'll consider which

2-categories such a presheaf might take values in. Only a little

familiarity with the notion of a category will be assumed!

Triply graded link homology (introduced by Khovanov and Rozansky) is a

categorification of the HOMFLYPT polynomial. In this talk I will discuss

recent joint work with Ben Webster which gives a geometric construction of this invariant in terms of equivariant constructible sheaves. In this

framework the Reidemeister moves have quite natural geometric proofs. A

generalisation of this construction yields a categorification of the

coloured HOMFLYPT polynomial, constructed (conjecturally) by Mackay, Stosic and Vaz. I will also describe how this approach leads to a natural formula for the Jones-Ocneanu trace in terms of the intersection cohomology of Schubert varieties in the special linear group.

Numerical computer codes implementing physics based models are the backbone of today's mechanical/aerospace engineering analysis and design methods. Such computational codes can be extremely expensive consisting of several millions of degrees of freedom. However, large models even with very detailed physics are often not enough to produce credible numerical results because of several types of uncertainties which exist in the whole process of physics based computational predictions. Such uncertainties include, but not limited to (a) parametric uncertainty (b) model inadequacy; (c) uncertain model calibration error coming from experiments and (d) computational uncertainty. These uncertainties must be assessed and systematically managed for credible computational predictions. This lecture will discuss a random matrix approach for addressing these issues in the context of complex structural dynamic systems. An asymptotic method based on eigenvalues and eigenvectors of Wishart random matrices will be discussed. Computational predictions will be validated against laboratory based experimental results.

The parabolic Anderson model is the Cauchy problem for the heat equation with random potential.  It offers a case study for the possible effects that a random, or irregular environment can have on a diffusion process.  In this talk I review results obtained for an extreme case of heavy-tailed potentials, among the effects we discuss our intermittency, strong localisation and ageing.